Rashba spin-orbit coupling in InSb nanowires under transverse electric field

Abstract

We investigate the Rashba spin-orbit coupling brought by transverse electric field in InSb nanowires. In small ${k}_{z}$ (${k}_{z}$ is the wave vector along the wire direction) range, the Rashba spin-orbit splitting energy has a linear relationship with ${k}_{z}$, so we can define a Rashba coefficient similarly to the quantum well case. We deduce some empirical formulas of the spin-orbit splitting energy and Rashba coefficient, and compare them with the effective-mass calculating results. It is interesting to find that the Rashba spin-orbit splitting energy decreases as ${k}_{z}$ increases when ${k}_{z}$ is large due to the ${k}_{z}$-quadratic term in the band energy. The Rashba coefficient increases with increasing electric field, and shows a saturating trend when the electric field is large. As the radius increases, the Rashba coefficient increases at first, then decreases. The effects of magnetic fields along different directions are discussed. The case where the magnetic field is along the wire direction or the electric field direction are similar. The spin state in an energy band changes smoothly as ${k}_{z}$ changes. The case where the magnetic field is perpendicular to the wire direction and the electric field direction is quite different from the above two cases, the ${k}_{z}$-positive and negative parts of the energy bands are not symmetrical, and the energy bands with different spins cross at a ${k}_{z}$-nonzero point, where the spin splitting energy and the effective $g$ factor are zero.